Hajek established a local asymptotic minimax risk bound for appropriate symmetric loss functions and also gave a necessary condition for the risk of an estimator to attain the lower bound. We extend these results to the case of asymmetric loss functions. The asymmetry brings about the shift of location of the loss functions. Besides, the optimal estimator that attains the bound is shown to have asymptotic normal distribution with asymptotic bias.
"Local Asymptotic Minimax Risk Bounds for Asymmetric Loss Functions." Ann. Statist. 22 (1) 39 - 48, March, 1994. https://doi.org/10.1214/aos/1176325356